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Abstract and Applied Analysis
Volume 2015, Article ID 413540, 14 pages
http://dx.doi.org/10.1155/2015/413540
Research Article

Integer and Fractional General -System and Its Application to Control Chaos and Synchronization

West University of Timisoara, 4 Bulevardul Vasile Pârvan, 300115 Timisoara, Romania

Received 23 October 2014; Revised 26 March 2015; Accepted 1 April 2015

Academic Editor: Huang Xia

Copyright © 2015 Mihaela Neamţu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose a three-dimensional autonomous nonlinear system, called the general system, which has potential applications in secure communications and the electronic circuit. For the general system with delayed feedback, regarding the delay as bifurcation parameter, we investigate the effect of the time delay on its dynamics. We determine conditions for the existence of the Hopf bifurcations and analyze their direction and stability. Also, the fractional order general -system is proposed and analyzed. We provide some numerical simulations, where the chaos attractor and the dynamics of the Lyapunov coefficients are taken into consideration. The effectiveness of the chaotic control and synchronization on schemes for the new fractional order chaotic system are verified by numerical simulations.

1. Introduction

Lorenz found the first canonical chaotic attractor [1]. During the time, it has been proved that chaos can occur in simple three-dimensional autonomous systems with one, two, and three nonlinearities. Tigan and Opris [2] proposed and analyzed a new chaotic three-dimensional nonlinear system, called system, which is similar to the Lorenz system. Because system allows a larger possibility in choosing the parameters of the system, it can display more complex dynamics [310].

Recently, based on the study of integer order chaos, the fractional order Lorenz system [11] and the fractional order Liu system [12] were introduced. The system with fractional order still shows the chaotic behavior [13, 14].

The system is described by [2]which is chaotic when , , and [2].

Li et al. [15] have proposed a new Lorenz-like chaotic system derived from (1). The nonlinear differential three-dimensional system is given bywhich is chaotic when , , and [15].

Chaotic phenomena in electric circuits have been studied with great interest. The electronic circuit for (2) is designed and a chaotic attractor is implemented and verified [15].

Yang [16] proposed another new Lorenz-like system. The nonlinear differential three-dimensional system iswhich is chaotic when , , , , , and [16].

Based on (1), (2), and (3), we propose a general system described by the following differential equations:where are positive real parameters, is a real parameter, and .

An experimental electronic circuit for (4) can be described and implemented in a similar way as in [15, 17].

System (4) is chaotic when , , , , , , and (see Figure 1).

Figure 1: Phase portrait of the chaotic attractor for the general system (4) with , , , , , , and .

The dynamics of Lyapunov exponents of system (4) is displayed in Figure 2.

Figure 2: Dynamics of Lyapunov exponents of (4) with , , , , , , and .

For , , , , and , system (4) is given by (1). For , , , , and , system (4) becomes (2). For , system (4) is (3).

Systems (1), (2), (3), and (4) are chaotic and using the method from [15] we can verify that they are not equivalent to the Lorenz, Chen, and Lü systems.

Time-delayed feedback is a powerful tool to control unstable periodic orbits or control unstable steady states [18]. Following the idea of Pyragas [19], as in [9, 18], we add a delayed force to the second equation of (4) and we obtain the delayed feedback control system:where is the time delay.

We assume that and , which indicates the strength of the feedback [10].

The fractional-order general system can be described bywhere is defined by [13]and is the derivative order that can be a complex number with the real part of . The numbers and are the limits of the operator. There are many definitions for general fractional derivative. The most frequently used ones are the Grunwald-Letnikov definition, the Riemann-Liouville, and the Caputo definitions.

As in [13], in this paper we use the Caputo definition for the fractional derivative.

In the present paper, we focus on (5) and (6). The aim is to provide a new investigation of the Hopf bifurcation and chaos control on the general system given by (5) and an analysis of the fractional general -system as well.

For system (5), we consider as the bifurcation parameter. When it passes through some certain critical values, the equilibrium will lose its stability and Hopf bifurcation will occur. We study the direction of the Hopf bifurcation, as well as the stability and period of the bifurcating periodic solutions. Moreover, with different values for and , we realize the chaos control.

The chaotic dynamics in the general -system with fractional derivative is taken into account. Some properties are given. Then, synchronization problem of (6) is provided.

The paper is structured as follows. Section 2 provides the stability of the steady states when there is no time delay. For system (5), Section 3 analyzes the local stability, the existence of the Hopf bifurcation, and the direction and the stability of the Hopf bifurcation. The analysis of system (6) is presented in Section 4. The chaos control of (6) is given in Section 5 and the synchronization in Section 6. Finally, conclusions are drawn in Section 7.

2. Existence of Steady States: Stability Analysis for System (4)

The equilibrium of system (4) can be obtained by solving the following algebraic system:

Then, we have the following.

Proposition 1. Consider the following:(i)If , then system (4) has only one real steady state .(ii)If , then system (4) has three real steady states: , , , where

In order to analyze the local stability of the above steady states, the Jacobian matrix of (4) is given byIf , the characteristic equation of is given byThe eigenvalues of (11) are , and .

For , both and have the same characteristic equation; that isThe eigenvalues of (12), which are dependent on parameters , can be obtained by the Cardano formula. Since are all positive real parameters, one can ensure that (12) has at least one eigenvalue with negative real part as . The other two eigenvalues could be(a)two negative real roots;(b)two positive real roots;(c)two complex-conjugate roots with negative real part;(d)two complex-conjugate roots with positive real part.

Therefore, analyzing the characteristic equation of (4) and using the Routh-Hurwitz theorem, we obtain the following propositions.

Proposition 2. Consider the following:(i)If , then the steady state of system (4) is locally asymptotically stable.(ii)If , then the steady state of system (4) is unstable.(iii)If and , wherethen the steady states and of system (4) are locally asymptotically stable (see Figures 3 and 4).(iv)If and , then the steady states and of system (4) are unstable (see Figures 5 and 6).

Figure 3: The steady state of system (4) is locally asymptotically stable when .
Figure 4: The steady state of system (4) is locally asymptotically stable when .
Figure 5: The steady state of system (4) is unstable when .
Figure 6: The steady state of system (4) is unstable when .

Using the Hopf bifurcation theorem [9, 16], we have the following.

Proposition 3. If and , then for the steady state of system (4), the corresponding characteristic equation has three eigenvalues: one negative and one pair of purely imaginary conjugate roots, satisfying ; that is, system (4) undergoes a Hopf bifurcation at the steady state (see Figures 7 and 8).

Figure 7: A Hopf bifurcation occurs at the steady state of system (4) when .
Figure 8: A Hopf bifurcation occurs at the steady state of system (4) when .

3. Local Stability and Hopf Bifurcation for System (5)

3.1. Local Stability and the Existence of Hopf Bifurcation for System (5)

Consider system (5). When , it becomes (4). Since the time delay does not change, the delayed feedback system (5) has the same equilibria as system (4).

From Proposition 1, under the assumption , system (5) also has three real steady states: , , and .

Now, we analyze the effect of the time delay on the stability of these steady states.

The linearization of system (5) at is

The characteristic equation of (14) is

Equation (15) has a negative root for all ; thus, we need to analyse the following equation:

If there is no delay, (16) becomes

Let be a root of (16). It follows thatwhich can be written aswhere

As in [9], we consider the following analysis:(1)If the conditionshold, then (19) has no positive roots.(2)If the conditionholds, then (19) has a unique positive root . Using (19), we have(3)If the conditionshold, then (19) has two positive roots . Thus, we have

A stability switch may occur, through the roots , where are given by (25). Therefore, from (18), we havewhere at which (16) has a pair of purely imaginary roots .

Considerthe root of (16) so that . Using (16) and considering , is given by

From (18) and (25), we obtain

Thus, if , we have

From the above findings we have the following.

Theorem 4. Let be defined by (26) and .(i)If holds, (16) and (17) have the same number of roots with positive real part for all .(ii)If either or holds, when ,  (16) and (17) have the same number of roots with positive real part. Moreover, if the transversality conditions (30) hold, then a Hopf bifurcation occurs at the steady state in (see Figure 9).

Figure 9: A Hopf bifurcation occurs at the steady state of system (5), when , .

From the symmetry of and , it is sufficient to analyze the stability of . By the linear transformationsystem (5) becomes

The characteristic equation of system (32) in iswhere

If there is no delay, (33) becomes

Let be a root of (33). Then, satisfies the equation

If , then (36) becomeswhere

Therefore, applying the findings from [20] we obtain the following.

Proposition 5. For the polynomial equation (37), we have the following:(i)If , then (37) has at least one positive root.(ii)If and , then (37) has no positive roots.(iii)If and , then (37) has positive roots if and only if and .

Suppose that (37) has positive roots. Without loss of generality, we assume that it has three positive roots, defined by , , and . Then, (36) has three positive roots , .

By direct computation, we have the following.

Theorem 6. For the simple pairs of conjugate purely imaginary roots of (36), , , we haveWe have , , with

For (33), using Proposition 5 and [21] to (33), we have the following.

Proposition 7 (see [10]). For (33) we have the following:(i)If and , then all roots with positive real part of (33) have the same sum to those of the polynomial equation (35) for all .(ii)If either or , , , and , then all roots with positive real parts of (33) have the same sum to those of the polynomial equation (35) for .

Let us denote the root of (33) by with .

Proposition 8. If and , then the transversality conditionis satisfied and and have the same signs.

Applying Propositions 7 and 8 to (33), we have the following theorems.

Theorem 9. Let and be defined by (39) and (40). Suppose that conditionholds.(1)If , then we have the following:(i)If and , then (33) has all the roots with negative real part for all and the steady state of system (5) is locally asymptotically stable.(ii)If or and , and , then (33) has all the roots with negative real part for all and the steady state of system (5) is locally asymptotically stable.(iii)If the conditions of (ii) hold and , then a Hopf bifurcation occurs at the steady state for .(2)If , then we have the following:(i)If and , then (33) has two roots with positive real part for all and steady state of system (5) is unstable.(ii)If or and , and , then (33) has two roots with positive real part for all and the steady state of system (5) is unstable.(iii)If the conditions of (ii) hold and , then a Hopf bifurcation occurs at the steady state for .

3.2. Direction and Stability of the Hopf Bifurcation

In the previous section, for system (5), we have obtained conditions for the Hopf bifurcations to occur for a sequence of values of . Using the techniques from normal form theory and center manifold theory introduced by [22], we determine the direction, the stability, and the periodicity of the bifurcating solutions. Let be the steady state of (5), where system (5) undergoes Hopf bifurcations at and are the corresponding pure imaginary roots of the characteristic equation.

For convenience, let , , and , , and dropping the bars for simplification of notation, system (5) can be written as a FDE (functional differential equation) in as follows:where and , are given, restrictively, by

From the previous section, if , system (5) undergoes Hopf bifurcations at . By Riesz representation theorem, there exists a function of bounded variation for so that

We can choosewhere is the Dirac delta function.

For we define

Then, system (43) is equivalent to the abstract differential equation:where , for .

For , we defineand the bilinear formwhere .

Then, and are adjoining operators. From Section 2, are eigenvalues of ; thus, they are also eigenvalues of .

By direct computation, we obtain thatwiththe eigenvector of corresponding to andwith the eigenvector of corresponding to , whereUsing the same notations as in [22], we compute the coordinates to describe the center manifold at .

Let be the solution of (43) when and define

On the center manifold we have where and and are local coordinates for center manifold in the direction of and . Note that is real if is real. We consider only real solutions. For the solution of (43), since , we have

We rewrite this equation aswhere

Note thatand .

We have

Using (61), we obtain

Since there are and in , we need to compute them.

From (43) and (56), we havewhere

Expending the above series and comparing the corresponding coefficients, we obtain

From (65), we know that, for ,

Comparing the coefficients with (66), we obtain

From (67) and (68) and the definition of , it follows that

Notice that andwhere is a constant vector.

In a similar way, we can obtainwhere is a constant vector.

From the definition of and (67), we obtainwhere .

By (65), we have

Substituting (71) and (75) into (73), we obtainwhich leads to

In a similar way, substituting (72) and (76) into (74), we obtain

Thus, we can determine and from (71) and (72). Furthermore, we can determine . Therefore, each in (64) is determined by the parameters and delay in (43). Thus, we can compute the following values:

The above quantities characterize the bifurcating periodic solutions in the center manifold at the critical value [22, 23]:(i) is the Hopf bifurcation of system (43).(ii) determines the direction of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solution exists for .(iii) determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if .(iv) determines the period of the bifurcating periodic solutions: the period increases (decreases) if .

4. Analysis of System (6)

Let . In this case, the fractional order system is commesurate-order [24].

Proposition 10. The initial value problem of the commensurate order system (6) can be rewritten as follows:where , ,Then, for constant , , fractional order system (6) has a unique solution, where is the Gamma function.

Proof. Consider the function , which is continuous and bounded on the interval , for any . For and , where , we can obtainwhere .
Therefore, the fractional-order general system meets the Lipschitz condition. Then, according to the existence and uniqueness theorem of the fractional-order system [25], the initial value problem of the commensurate order system (81) has a unique solution in the interval .
System (6) is dissipative for , becausewhere . Meanwhile, it is convergent in an exponential rate That is to say, an initial volume is the volume element at time contraction of volume element . When all the trajectory of systems will eventually be restricted in a volume element for zero point sets, and incremental dynamic behavior of it will be fixed in an attractor.

In order to determine the stability conditions for the steady states and , we first consider the integer-order case. Based on Proposition 2, and are locally asymptotically stable if and only if , whereand . It is known that the fractional-order system is at least as stable as their integer-order counterpart, so we have the following conclusion.

Proposition 11. The steady states and are stable with for .

When , the steady states and become unstable in an integer-order system but may be stable in the fractional-order case.

5. Chaos Control of System (6)

In this section, we want to control the chaos for the fractional-order general system (6) to the steady state denoted by via feedback control.

An -dimensional fractional-order system can be described as [13]where , .

The system with controller is given by [13]where is the matrix of positive feedback gains and is the steady state of system (87).

The controlled fractional-order general system (6) iswhere , , are the external control inputs. The control law of single state variables feedback has the following form [13]:where is the matrix of positive feedback gains and is the steady state of system (6).

The characteristic equation of the controlled system (89) evaluated at the steady state is

The fractional-order Routh-Hurwitz conditions lead to

6. Synchronization of System (6)

The nonlinear control method is used to design control in order to make the drive system (6) and response system state synchronization [14]. Two identical systems are introduced, one is the drive system and the other system added a nonlinear control to be the response system. Corresponding to (6), the drive system isand the response system iswhere is the nonlinear synchronization controller. The drive system and the response system achieve synchronization under the driver of . From (93) and (94), the error system is obtained:where . We have to find the proper control function , , so that the response system (94) globally synchronizes with drive system (93); that is, , where . According to [14], we propose the following control law for system (94):where , , and are the control parameters.

Proposition 12 (see [14]). For any initial conditions, if , then the drive system and response system will synchronize.

7. Numerical Simulations

Now, we illustrate the findings from the previous sections. We have proved that at some critical values of the delay, a family of periodic solutions bifurcate from the steady states of system (5) and the stability of the steady states may be changed.

The numerical simulations indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.

We consider the delayed feedback control system (5) in the following particular form:which has three steady states , , and . When and , system (97) is chaotic.

The characteristic equation of system (97) is given by

If there is no delay, (98) has a negative root and a pair of complex roots with positive real part.

Using Theorem 9, we have the following:(i)For , , , (98) has two roots with positive real part for all , and the steady state (or ) of system (97) is unstable.(ii)For chaos can occur.

In what follows, we consider . In this case: , ,

Then, .(i)For , the steady states and of system (97) are unstable.(ii)For , the steady states and of system (97) are locally asymptotically stable.(iii)For , for , , system (97) undergos a Hopf bifurcation at the steady states and .

For values of , which satisfy the above conditions, we obtain the dynamical behaviors in Figures 10, 11, 12, and 13.

Figure 10: For system (97), there is chaos when , .
Figure 11: For system (97), there is chaos when , .
Figure 12: For system (97), an unstable periodic solution bifurcates from the steady state when , .
Figure 13: For system (97), an unstable periodic solution bifurcates from the steady state when , .

For the numerical simulation of the fractional differential equations (FDE) (6), with , , , , , , and , we use the method from [26, 27]. We obtain Figures 14, 15, and 16.

Figure 14: Asymptotic stable solution for the fractional general system (6) when .
Figure 15: Periodic solution for the fractional general system (6) when .
Figure 16: Chaos occurs for the fractional general system (6) when .

For the controlled fractional order general -system (89), using the matrix of positive feedback gains, we obtain Figure 17.

Figure 17: Asymptotic stable solution for the controlled fractional general system (89), when .

8. Conclusion

In the present paper we introduce a generalized -system where the time delay is present. The linear stability is analyzed by using the Routh Hurwitz criterion. The existence of the Hopf bifurcation is studied. Then, the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Chaotic behavior is also taken into account. The numerical simulations show that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a periodic orbit.

Furthermore, the fractional-order general system has been proposed. The dynamics, chaos control as well as synchronization have been investigated.

The present study will be continued for the system which describes the financial risk [28]. Also, as in [29], the fractional-order chaotic complex system will be taken into consideration.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank professor Dumitru Opris for useful conversations on the topics of this paper. The authors would like to express their gratitude to the editor and the anonymous referees for the comments on this paper.

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